Possible, yes. Economically efficient? Never. Would you like the results? Hell no.

I'm an ex-lab manager who worked in a portrait lab through the transition from film to digital photography. So I know that all the silver halide paper manufacturers evolved their papers to optimize them for short exposure in digital devices such as LED printers, laser printers, light valve printers, etc. In the lab I worked for, we had 40 such devices — various Noritsu mini-labs — in 2008. Each was capable of at least 400, to as many as 600 8x10 inch prints per hour.

While the current silver halide papers will work with old optical printing processes, you wouldn't like the results as much as you would have 25 years ago. Kodak started optimizing their paper for digital printers in the late 1990s.

In any case, using even a 4K digital projector would not provide sufficient resolution to do justice to a digital file. It's optimized for positive image projection, not printing. Control over color would be nearly impossible, since you need to convert a positive image to a negative and add an orange filter to make it look — to the paper — like a color negative. Fog from stray light coming from the fan vents would be another issue. Exposure control? Fuggeddaboudit. At the end of the day, printing from a digital projector would be, in most respects, similar to printing from an enlarger without control over color and exposure.

The finest prints made today are printed from applications like Adobe Lightroom, directly to high-end Canon or Epson inkjet photo printers. Their archival properties are three to five times those of silver halide papers, and their color gamuts (ranges of possible color saturation) are far more appealing, too. Even a 44" Epson is under $6000, is completely stable, and works the first time out of the box, provided you understand color management. (We had three in our lab.) And it works in normal room lighting.

I think it would work. You need to work out a way to totally stop light from the projector and somehow control the exposure on the paper. My last foray into color darkroom used paper that required a few seconds to expose, but not more than 10. And because it was color paper (if I remember correctly from over 40 years ago) you could not have a "safelight" as in the black & white dark room. That means that the projector, which needs to be ventilated and spews light out from more than just the lens, would have to be some how "outside" the room.

I think it is doable, but will be a challenge. Let us know how it works out!